A Min, + System Theory for Constrained Traffic Regulation and Dynamic Service Guarantees

Transcription

1 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 10, NO 6, DECEMBER A Min, + System Theory for Constrained Traffic Regulation Dynamic Service Guarantees Cheng-Shang Chang, Senior Member, IEEE, Rene L Cruz, Senior Member, IEEE, Jean-Yves Le Boudec, Patrick Thiran, Member, IEEE Abstract By extending the system theory under the (min, +) algebra to the time-varying setting, we solve the problem of constrained traffic regulation develop a calculus for dynamic service guarantees For a constrained traffic-regulation problem with maximum tolerable delay maximum buffer size, the optimal regulator that generates the output traffic conforming to a subadditive envelope minimizes the number of discarded packets is a concatenation of the -clipper with ( ) = min[ ( + ) ( ) + ] the maximal -regulator The -clipper is a bufferless device, which optimally drops packets as necessary in order that its output be conformant to an envelope The maximal -regulator is a buffered device that delays packets as necessary in order that its output be conformant to an envelope The maximal -regulator is a linear time-invariant filter with impulse response, under the (min +) algebra To provide dynamic service guarantees in a network, we develop the concept of a dynamic server as a basic network element Dynamic servers can be joined by concatenation, filter bank summation, feedback to form a composite dynamic server We also show that dynamic service guarantees for multiple input streams sharing a work-conserving link can be achieved by a dynamic service curve earliest deadline scheduling algorithm, if an appropriate admission control is enforced Index Terms Buffer overflow, (min +) algebra, network calculus, packet losses, performance analysis, traffic shaping I INTRODUCTION FUTURE high-speed digital networks aim to provide integrated services, including voice, video, fax, data To control interaction among traffic generated by different sources, traffic regulation seems inevitable In [10], Cruz proposed the following deterministic traffic characterization A traffic stream, described by a nondecreasing sequence Manuscript received October 15, 2001; revised January 30, 2002; approved by IEEE/ACM TRANSACTIONS ON NETWORKING Editor M Ammar This work was supported in part by the National Science Council, Taiwan, ROC, under Contract NSC E Contract NSC E , by the US National Science Foundation under Grant NCR Grant NCR , by the Center for Wireless Communications at the University of California at San Diego, by the Priority Program in Information Communication Structures of the Swiss National Science under Grant SPP ICS C-S Chang is with the Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan, ROC ( cschang@eenthuedutw) R L Cruz is with the Department of Electrical Computer Engineering, University of California at San Diego, La Jolla, CA USA ( cruz@eceucsdedu) J-Y Le Boudec P Thiran are with the School of Computer Communication Sciences, Ecole Polytechnique Fédérale de Lausanne, Luasanne CH-1015, Switerl ( jean-yvesleboudec@epflch) Digital Object Identifier /TNET function [with, called an envelope, if ], conforms to a Without loss of generality, an envelope can be assumed to be subadditive [6], ie, for all Using this characterization, a calculus is developed in [10] [11] to compute deterministic performance measures, such as bounds on delay bounds on queue length Traffic regulation addresses the problem of modifying a traffic stream so that it conforms to a subadditive envelope The problem of traffic regulation was treated systematically in [8] [20], where it is shown that the optimal traffic regulator that generates an output conforming to a subadditive envelope for an input is a linear time-invariant filter with the impulse response algebra, ie, under the We call such a filter the maximal -regulator This characterization was also observed in [1], [2], [27] As the buffer in the maximal -regulator is assumed to be infinite, packets from the input might be queued at the regulator For a real-time service, the delay of a queued packet at the regulator might exceed a maximum tolerable delay such a packet should be discarded (ie, clipped) The problem of traffic regulation with such a delay constraint is called the constrained traffic-regulation problem in [19] Its objective is to find a regulator that not only generates traffic conforming to an envelope, but also minimizes the number of discarded packets In addition to the delay constraint, Konstantopoulos Anantharam [19] also considered the buffer constraint for the regulator For, they derived optimal traffic regulators that satisfied either the delay constraint or the buffer constraint Cruz Taneja [16] considered the zero delay case of the constrained traffic-regulation problem This is also the case without any buffer By extending the time-invariant filtering theory under the algebra to the time-varying setting, it is shown there that the departure process of the optimal zero-delay regulator, which generates a departure process conformant to, is the subadditive closure [8] of the arrival process convolved with Such a bufferless regulator is called the -clipper in [16] Motivated by all these works, one of the main objectives of this paper is to provide an optimal implementable solution for the general constrained traffic-regulation problem with both /02$ IEEE

2 806 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 10, NO 6, DECEMBER 2002 the delay buffer constraints As in [16], our approach is based on the time-varying filtering theory under the algebra By extending the subadditive closure in [8] to the timevarying setting, we show that the -clipper with input output can be implemented using the following recursive equation: The computation complexity of the -clipper is almost the same as that of the maximal -regulator The recursive equation also implies that the -clipper is greedy Packets are discarded only when needed For the constrained traffic-regulation problem with maximum tolerable delay maximum buffer size, the optimal traffic regulator is shown to be a concatenation of the -clipper with the maximal -regulator The solution is intuitive as the output from the -clipper conforms to the envelope that yields bounded delay bounded queue length at the maximal -regulator For example, when, the corresponding -clipper can be implemented by parallel bufferless -leaky buckets A packet is discarded if it cannot be admitted to one of these leaky buckets The output from the -clipper is then fed into parallel -leaky buckets In addition, the time-varying filtering theory can also be used for dynamic service guarantees By extending the concept of the service curve in [1], [12], [20] to a bivariate function, we define a dynamic -server for an input if its output satisfies Analogous to the time-invariant filtering theory in [1], [8], [20], a dynamic -server can be viewed as a linear filter with the time-varying impulse response It can be combined by concatenation, filter bank summation, feedback to form a composite dynamic server We illustrate the use of the dynamic server by considering a work-conserving link with a timevarying capacity a dynamic window-flow-control problem We also show that dynamic service guarantees for multiple input streams sharing a work-conserving link can be achieved by a dynamic service curve earliest deadline (SCED) scheduling algorithm if an appropriate admission control is enforced As the SCED algorithm in [27], the dynamic SCED algorithm is an earliest deadline first (EDF) policy that schedules packets according to their deadlines The remainder of this paper is organized as follows In Section II, we introduce the time-varying filtering theory under the algebra The development is parallel to the time-invariant filtering theory in [1], [8], [20] The reader is also referred to [3] [4], which contain results overlapping with this paper In Sections III IV, we introduce the maximal dynamic traffic regulators maximal dynamic clippers, respectively These are used for solving the problem of constrained traffic regulation in Section V In Section VI, we develop the concept of dynamic servers their associated calculus We show in Section VII that the dynamic SCED algorithm can be used to achieve dynamic service guarantees We conclude the paper in Section VIII by discussing possible extensions applications II TIME-VARYING FILTERING THEORY UNDER THE MIN, ALGEBRA In the section, we introduce the time-varying filtering theory under the algebra The development is parallel to the time-invariant filtering theory in [1], [8], [20] To extend the algebra to the time-varying setting, we consider the family of bivariate functions for all Thus, for any, is nonnegative nondecreasing in For any two bivariate functions in, we say (respectively, )if [respectively, ] for all We define the following two operations for functions in i) ( ) the pointwise minimum of two functions ii) (convolution) the convolution of two functions under the algebra One can easily verify that is a complete dioid (see, eg, [5]) with the zero function the identity function, where for all, if otherwise To be precise, we have the following properties 1 (Associativity) 2 (Commutativity) 3 (Distributivity for infinite sums ) For any two sequences of functions in 4 (Zero element) 5 (Absorbing zero element) 6 (Identity element)

3 CHANG et al: MIN, SYSTEM THEORY FOR CONSTRAINED TRAFFIC REGULATION AND DYNAMIC SERVICE GUARANTEES (Idempotency of addition) Thus, The key difference to the time-invariant filtering theory is that we do not have the commutative property for in, ie, in general Let : That is, a function if for all As in the time-invariant case, we still have the following monotonicity 8 (Monotonicity) If (respectively, ) isin, then (respectively, ) If both are in, then For any function, define the unitary operator (called the closure operation in this paper) where is the self-convolution of for times, ie,,, The limit in (1) exists as is decreasing in Exping (1) yields where is any subset of with In addition to the algebraic properties, we present several important properties in Lemmas that will be used to prove results for constrained traffic regulation service guarantees Lemma 21: Suppose that i) (Monotonicity) If, then ii) (Closure properties) iii) (Maximum solution) is the maximum solution of the equation, ie, for any satisfying, iv) can be computed recursively from the following equations: (1) (2) Lemma 22 (Feedback): Suppose that i) For the equation is the maximum solution ii) If, then is the unique solution iii) Under the condition in ii), if then The proofs for Lemma 22 are identical to those in [8] [9], thus, are omitted Remark 23: As in [8], let,, be the set of nonnegative nondecreasing functions Also, let be the subset of functions in with One may then define the convolution of a function a bivariate function as follows: Under such a definition, is in One may view as a special case of for some with for all, for all Thus, the results in Lemma 22 still hold Remark 24: A bivariate function is time invariant if By letting, one can easily verify that is time invariant if only if there exists some such that As a result, time-invariant bivariate functions commute To see this, consider two invariant functions let Then (4) An important corollary of (4) is that Lemma 21(v) can be simplified as follows (cf [8, Lemma 22(xi)]): Remark 25: A bivariate function is additive if (3) (5) v) Proof: As the proofs for i) iv) are identical to those in [8] [9], we only prove v) From the monotonicity, Thus, On the other h, one has Thus, Similarly, This implies For an additive bivariate function, one easily check that which implies that Note that a bivariate function is additive if only if there is a function such that This can be easily verified by choosing

4 808 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 10, NO 6, DECEMBER 2002 III DYNAMIC TRAFFIC REGULATION Given a sequence, it is defined in [10] [11] that conforms to the (static) upper envelope if for all It is also shown in [8] [1] that the optimal traffic regulator that generates output traffic conforming to a subadditive envelope is a linear time-invariant filter with the impulse response under the algebra In this section, we extend such a result to the time-varying setting We start from extending the definition of a static envelope to a dynamic envelope Definition 31: A sequence is said to conform to the dynamic upper envelope if for all there holds As in [8] [9], this characterization has the following equivalent statements The proof is omitted Lemma 32: Suppose that The following statements are equivalent i) conforms to the dynamic upper envelope ii) iii) iv) conforms to the dynamic upper envelope Given a dynamic upper envelope, one can construct a regulator such that, for any input, the output from the regulator conforms to the dynamic upper envelope This is done in the following theorem Once again, the proof is omitted Theorem 33: Suppose that Let i) (Traffic regulation) conforms to the dynamic upper envelope, thus, also conforms to the dynamic upper envelope ii) (Flow constraint) iii) (Optimality) For any that satisfies i) ii), one has iv) (Conformity) conforms to the dynamic upper envelope if only if The construction is called the maximal dynamic -regulator (for the input ) As in the time-invariant case, the flow constraint corresponds to one of the causal conditions in [19] as the number of departures cannot be larger than the number of arrivals Theorem 33(iii) shows that, under the flow constraint the constraint that the output traffic conforms to the dynamic upper envelope, the maximal -regulator is the best construction that one can implement Example 34 (Work-Conserving Link With a Time-Varying Capacity): Consider a work-conserving link with a time-varying capacity Let be the maximum number of packets that can be served at time, be the cumulative capacity in the interval, be the cumulative capacity in the interval Let be the input the output from the work-conserving link Denote by the number of packets at the link at time The work-conserving link is then governed by Lindley s equation (6) where Suppose Recursive expansion of Lindley s equation yields Since,wehave As is an additive bivariate function, we have from Remark 25 that, which shows that the work-conserving link is the maximal dynamic -regulator This example also shows that the calculation of the convolution in (8) can be easily implemented by the recursion in (6) We note that a work-conserving link with a time-varying capacity is also equivalent to a time-varying (greedy) shaper in [20] Example 35 (Traffic Regulation With a Capacity Constraint): Consider a link with a time-varying capacity The link is not necessarily work conserving As in the previous example, let be the maximum number of packets that can be served at time, be the cumulative capacity by time Let be the input output from the link Though the link may not be work conserving, the output is still constrained by the capacity, ie, Suppose that we would like to perform traffic regulation for the input such that the output conforms to the static envelope, ie, (7) (8) (9) (10) From Theorem 33, we know that the optimal implementation for the output to satisfy (9) (10) is the maximal dynamic -regulator with (11) If is bounded above by if the cumulative time-varying capacity is bounded below by some curve over any time window, ie, if for all,, then one can derive static service curves bounding below the maximal dynamic -regulator (11) Such curves are obtained in [15], [22], [23] IV DYNAMIC TRAFFIC CLIPPING The maximal dynamic -regulator solves the traffic-regulation problem with an infinite buffer In this section, we consider the traffic-regulation problem without a buffer The question is then how one drops packets optimally such that the output conforms to a dynamic envelope Such a problem was previously solved in [16]; however, the solution in [16] cannot be easily implemented directly In the following theorem, we present a recursive construction for the solution Theorem 41: Suppose that Let, where The following statements then hold

5 CHANG et al: MIN, SYSTEM THEORY FOR CONSTRAINED TRAFFIC REGULATION AND DYNAMIC SERVICE GUARANTEES 809 i) (Traffic regulation) conforms to the dynamic upper envelope ii) (Clipping constraint) for all iii) (Optimality) For any that satisfies i) ii), one has iv) can be constructed by the following recursive equation: It then follows from Lemma 22(i) that is the maximum solution of (16) Note that Thus, is the maximum solution that satisfies i) ii) To see iv), note from Lemma 21(iv) that can be constructed recursively as follows: (12) with v) (Conformity) conforms to the dynamic upper envelope if only if The construction in (12) is called the maximal dynamic -clipper (for the input ) in this paper Proof: For any, we have from Lemma 21(iv) that with Since satisfies the clipping constraint, (17) This implies that, hence, so that is conformant to, establishing i) To see ii), note similarly that Thus, (18) Next, we establish iii) Suppose that ii) Since satisfies i) (13) To prove v), note that if, then it follows from (18) that Thus, conforms to the dynamic envelope On the other h, if conforms to the dynamic envelope, then where for As conforms to the dynamic envelope This implies As from Remark 25 (14) The inequality in the clipping constraint in ii) is equivalent to for all it can be rewritten as (15) with The constraints in (13) (15) are equivalent to We note that the original representation in [16] is that This is equivalent to our result in Theorem 41, as can be seen from Lemma 21(v), the fact that As in Lemma 21(v), one also has the following equivalent implementation: Applying the distributivity the fact that yields (16) (19) Note that the key difference between Theorems is the clipping constraint The clipping constraint implies that, in any given slot, the packets departing are a subset of the packets arriving in the same slot Let

6 810 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 10, NO 6, DECEMBER 2002 we have be the number of packets clipped at time From (12), (20) Observe from (20) that packet loss occurs at time only when at least one of the following inequalities is violated: (21) When this happens, one then discards packets to the extent so that the above inequalities are all satisfied Note also that (12) implies that the maximal dynamic -clipper can be implemented in real time since the value of depends only on for In the following example, we illustrate how one implements the maximal dynamic -clipper by a work-conserving link with a finite buffer when for Example 42 (Work-Conserving Link With a Finite Buffer): Consider the work-conserving link with a time-varying capacity in Example 34 In addition, we assume that the buffer size of the link is, ie, at most, packets can be stored at the link Packets that arrive at the link find the buffer full are lost As in Example 34, let be the input output from the work-conserving link Denote by the number of packets at the link at time We then need to modify Lindley s equation in (6) as follows: (22) The number of lost packets at time, denoted by, is then Let be the effective input to the link, ie, For the effective input, the work-conserving link behaves like a work-conserving link with an infinite buffer Thus, we have from (7) that (23) In view of (20), the effective input to the work-conserving link with a finite buffer is, in fact, the output of the maximal dynamic -clipper with, In particular, when for all, we can implement the maximal dynamic -clipper with by constructing the effective input of a work-conserving link with constant capacity buffer This example also shows that a direct calculation of the convolution in (12) may not be necessary, the convolution in this example can be computed recursively by (22) For the maximal dynamic -clipper with the input the output, let be the cumulative losses at the clipper by time As in Theorem 41, using (2) yields (24) where is any subset of with This was previously shown in [16, Corollary 1] A similar result is also obtained in [22] for both the continuous discrete time settings Example 43 (Clippers in Tem): Now we compare the output from the maximal dynamic -clipper a concatenation of the maximal dynamic -clipper the maximal dynamic -clipper Let be the input to both systems, be the output from the maximal dynamic -clipper, be the output from the maximal dynamic -clipper, be the output from the maximal dynamic -clipper Also let be the cumulative losses at the maximal dynamic -clipper by time Similarly, let From Theorem 41, we have for all assuming This then implies This implies that conforms to the dynamic upper envelope that Thus, for all by Theorem 41 In fact, a concatenation of the maximal dynamic -clipper maximal dynamic -clipper is a suboptimal implementation of an -clipper The reason for this, as observed in [16], is that the discarding of packets in the -clipper is not accounted for in the -clipper

7 CHANG et al: MIN, SYSTEM THEORY FOR CONSTRAINED TRAFFIC REGULATION AND DYNAMIC SERVICE GUARANTEES 811 Example 44 (Clippers in Parallel): Continue from the previous example Since clippers in tem are suboptimal may yield more cumulative losses than the optimal one, we may use this to compare the cumulative losses for clippers in parallel Now suppose both the maximal dynamic -clipper maximal dynamic -clipper are fed with the input Let be the outputs from these two clippers be the cumulative losses at these two clippers by time Clearly, It is easy to see from (24) that Thus, we still have for all This is previously reported in [16, Corollary 2] V CONSTRAINED TRAFFIC REGULATION The two traffic-regulation problems, with an infinite buffer without a buffer, are two extreme cases In practice, packets (or cells) may be queued delayed at a regulator However, there might be constraints for the buffer size the delay In this regard, one might have to discard (ie, clip) some packets from the input so that the buffer delay constraints can be satisfied The question is then how one discard packets optimally so that the number of clipped packets can be minimized Such a problem is called constrained traffic regulation was first considered in [19] for -leaky buckets Our objective in this section is to provide a general, simple, optimal solution for the constrained traffic-regulation problem To formalize the problem of constrained traffic regulation with buffer delay constraints, we let be the input be the output from the regulator We require that the buffer occupancy in the regulator be less than or equal to, the delay be bounded above by, that the output be conformant to a dynamic envelope Due to these constraints, packets may need to be discarded Let be the effective input, ie, counts the total number of packets arriving up to including slot, which eventually depart the regulator without being discarded The objective is to maximize the effective input the output, given the buffer delay constraints the constraint that conforms to the dynamic envelope More formally, given the input a dynamic envelope, we seek, which are as large as possible subject to the following constraints (C1) (Clipping constraint) for all (C2) (Buffer constraint) for all, where is the buffer size at the regulator (C3) (Delay constraint) for all, where is the maximum tolerable delay at the regulator (as the regulator serves packets in the FCFS order) (C4) (Traffic regulation) conforms to the dynamic upper envelope (C5) (Flow constraint) for all The clipping constraint implies that the packets in the effective input is a subset of the packets in for any time We note that the clipping constraint does not imply that packets arriving at time have to be clipped at time In fact, they could be clipped at some time later than However, as will be shown below, optimal clipping can be greedy only those packets arriving at time need to be clipped at time Note also that the natural buffer constraint should be, where is the cumulative number of packets arriving up to time, which have not been discarded at the end of slot Our buffer constraint is, in fact, less restrictive as for all However, as the below theorem shows, the optimal value of can be computed without knowledge of for so that packets that will eventually be discarded in an optimal clipper can, in fact, be discarded when they arrive Assuming this is the case, the backlog of packets in the optimal regulator at the end of slot is Theorem 51: Suppose that Let be the output from the maximal dynamic -clipper for the input, where Also, let be the output from the maximal dynamic -regulator for the input All constraints (C1) (C5) are then satisfied Moreover, for any that satisfy (C1) (C5), one has The construction of, based on a concatenation of the maximal dynamic -clipper the maximal dynamic -regulator, is called the maximal dynamic -regulator with delay buffer Proof: Suppose that are as stated in the theorem Theorem 41 then implies (C1), also implies that Conditions (C4) (C5) follow from Theorem 33(i) (ii) To establish (C2), note that Similarly, to establish (C3), note that Since for is nonnegative, it, therefore, follows that, which establishes (C3) Thus, (C1) (C5) are satisfied as claimed Next, suppose that satisfy (C1) (C5) From Theorem 33(iii), we know that, under the flow constraint in (C5) the traffic constraint (C4), we have (25) Moreover, combining this with (C2) (C3), we obtain (C2 ) (Buffer constraint) for all (C3 ) (Delay constraint) for all The buffer constraint in (C2 ) can be rewritten as (26)

8 812 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 10, NO 6, DECEMBER 2002 with Since is nondecreasing in is nonnegative (27) Due to the conditions in (27), the delay constraint in (C3 ) can be rewritten as (28) with Using the idempotency distributivity, the constraints in (26) (28) are equivalent to (29) Note that for all, where is defined in Theorem 51 Thus, conforms to the dynamic envelope Using Theorem 41(iii) the assumption that satisfies (C1), it, therefore, follows that From the monotonicity of, we also have from (25) that Fig 1 Work-conserving link with a finite buffer There is a well-known duality interpretation for a work-conserving link with a finite buffer One may view the cumulative capacity as the cumulative number of tokens generated by time As in a leaky bucket, every packet needs to grab a token for its departure Thus, packet losses occur when the buffer is full token losses occur when the buffer is empty To be precise, let be the number of packets at the link at time, be the cumulative number of packet losses by time, be the cumulative number of token losses by time Fig 1 represents this system One then has the following conditions of complementary slackness: for all for all where if the event is true 0 otherwise As We note that, for the special cases that (without delay constraint) that (without buffer constraint), the results were previous obtained in [22, Ch 9] The result in Theorem 51 not only finds a representation of the optimal traffic regulator that satisfies both the delay buffer constraints, but also provides a method for the implementation of such a regulator In [19], the buffer delay constraints are treated separately, it is shown that the optimal solution can be implemented by the greedy flow controller, which discards packets only when needed As shown in Theorem 51, the maximal dynamic -regulator with delay buffer is still the greedy flow controller as the maximal dynamic -clipper discards packets only when needed Example 52 (Work-Conserving Link with a Finite Buffer): In this example, we show that a work-conserving link with a finite buffer solves a traffic-regulation problem with a buffer constraint Consider the work-conserving link with a time-varying capacity a finite buffer in Example 42 As in Examples 34 42, let,, be the input, effective input, output of the link, respectively As we have shown from Example 42, the effective input to the link is, in fact, the output of the maximal dynamic -clipper with Also, from Example 34, the output from the link is the output from the maximal dynamic -regulator with Thus, the link is a concatenation of the maximal dynamic -clipper the maximal dynamic -regulator We then have from Theorem 51 that the work-conserving link with a finite buffer is the maximal dynamic -regulator with buffer, where the work-conserving link with a finite buffer solves the so-called Skorokhod reflection problem with two boundaries [28], where is the free process, is the lower boundary process, is the upper boundary process (see, eg, [18] [19] for more detailed discussions of the reflection problem) Since the work-conserving link with a finite buffer also solves the buffer-constrained traffic-regulation problem, it follows from (24) that the upper boundary process of the reflection problem admits the following close-form representation (in terms of the free process): where is any subset of with Using (2), one can also show that the lower boundary process admits the following closed-form representation: where the sum in the right-h side is 0 for We also note the queue-length process can also be represented in closed form Two representations based on, plus operations were given in [14]

9 CHANG et al: MIN, SYSTEM THEORY FOR CONSTRAINED TRAFFIC REGULATION AND DYNAMIC SERVICE GUARANTEES 813 Example 53 (Multiple Leaky Buckets With Buffer Delay Constraints): Now consider the maximal dynamic -regulator with delay buffer when This corresponds to the case of multiple leaky buckets with the delay constraint the buffer constraint In this case, Thus, one can construct the maximal dynamic -clipper by feeding the input to parallel bufferless, -leaky buckets A packet is discarded (or clipped) if it cannot be admitted to one of these leaky buckets The output from the maximal dynamic -clipper is then fed into another parallel -leaky buckets with buffer This example also shows that a direct calculation of the convolution in (12) may not be necessary, as leaky buckets are known to have recursive implementations To bound the cumulative loss for the maximal dynamic -clipper in this example, we may apply the comparison result in Example 44 Consider maximal dynamic clippers, all subject to the same input The th clipper is the maximal dynamic -clipper with Let be the cumulative number of losses by time at the th clipper From Example 44, is an upper bound for the cumulative loss for the maximal dynamic -clipper Now is much easier to compute, as it is simply the cumulative loss for a work-conserving link with capacity buffer in Example 52 Example 54 (Bounding Losses by Segregation Between Buffer Policer): We have shown in Theorem 51 that the maximal dynamic -regulator with buffer is the optimal implementation to generate an output conforming to the dynamic envelope subject to the buffer constraint In this example, we will show that segregation of buffer discard policing discard provides an upper bound on the cumulative losses for the maximal dynamic -regulator with buffer As we have shown in Theorem 51, the first stage of the maximal dynamic -regulator with buffer is the maximal dynamic -clipper, where (30) Let,, be its input, output, the cumulative losses by time, ie, We now compare the cumulative losses with the losses in another system made of two parts, as shown in Fig 2 The first part is some causal system with storage capacity We know, however, that the first part discards packets as soon as the total backlogged packets in this system exceeds This operation is called buffer discard, Fig 2 Storage/policer system with separation between losses due to buffer discard policing discard the amount of buffer discarded packets by time is denoted by The second part is the maximal dynamic -clipper referred to here as the policer Packets are discarded as soon as the total output of the storage system exceeds the maximum output allowed by the policer This operation is called policing discard, the amount of discarded packets by time due to policing is denoted by We show that Let be the output of the buffer clipper, be the input output of the policer clipper, respectively As is the output of the maximal dynamic -clipper Now let be the effective input to the system, ie, Also, as shown in Fig 2, we have Since have from (32) that (31) (32) (33) (34) is a nondecreasing function in,we (35) On the other h, because the storage system is causal, it satisfies the flow constraint Since its storage space is limited to Using (32) (33), we have for all, we also have From (30), (31), (34), (36), (37), it then follows that (36) (37) (38) Combining (35) with (38), one notices that satisfies the same constraints as As is the output from the optimal implemen-

10 814 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 10, NO 6, DECEMBER 2002 tation in Theorem 51, it follows that or, equivalently, that Such a separation of resources between the buffered system policing system is used in the estimation of loss probability for devising statistical CAC algorithms as proposed by Lo Presti et al [26] (see also Elwalid et al [17]) VI DYNAMIC SERVICE GUARANTEES To guarantee end-to-end deterministic QoS for an input, the concept of service curves is developed in [1], [12], [20] to work with the static envelopes of Cruz [10], [11] A server is called a static -server ( ) for an input sequence if its output sequence satisfies (39) for all Based on this, there is an associated filtering theory (under the algebra) in [1], [8], [20] that eases design computation of deterministic QoS Our main objective in this section is to extend the concept of service curves the associated filtering theory to the time-varying setting so that dynamic QoS can be guaranteed The theory is based on the following definition of a dynamic -server Definition 61 (Dynamic -Server): A server is called a dynamic -server ( ) for an input sequence if its output sequence satisfies, ie, (40) for all If the inequality in (40) is satisfied for all input sequences, then we say the dynamic -server is universal If the inequality in (40) is an equality, we say the dynamic -server is exact Analogous to the filtering theory for static service curves, one may view the right-h side of (40) as the output from a linear filter with the time-varying impulse response under the algebra If is a time-invariant bivariate function, then the dynamic -server is equivalent to a static -server, where Clearly, the maximal dynamic -regulator is a universal exact dynamic -server Analogous to the time-invariant case, one has the following properties in Theorems for dynamic -servers The proofs are omitted, as they are identical to those in [8] [9] Theorem 62 (Concatenation): A concatenation of a dynamic -server for an input sequence a dynamic -server for the output from the dynamic -server is a dynamic -server for, where Theorem 63 (Filter Bank Summation): Consider an input sequence Let (respectively, ) be the output from a dynamic -server (respectively, -server) for The output from the filter bank summation, denoted by, is The filter bank summation of a dynamic -server for a dynamic -server for is then a dynamic -server for, where Theorem 64 (Feedback): Consider an input sequence a dynamic -server for, where, is the output from the dynamic -server If, then the feedback system is a dynamic -server for Theorem 65: Consider a dynamic -server for Let be the output Also, let be the maximum queue length at the server, where Let : for all be the maximum delay at the server Suppose that conforms to the dynamic upper envelope i) (Queue length) ii) (Output burstiness) If, then conforms to the dynamic upper envelope, where iii) (Delay) : Remark 66: As the maximal dynamic -regulator is a dynamic -server, there is an intuitive explanation why the maximal dynamic -regulator with delay buffer is a concatenation of the maximal dynamic -clipper (with being defined in Theorem 51) the maximal dynamic -regulator As shown in Theorem 41, the output from the maximal dynamic -clipper conforms to the dynamic envelope When such an output is fed to the maximal dynamic -regulator, one has from Theorem 65 that the delay at the maximal dynamic -regulator is bounded above by the queue length is also bounded above by Thus, both the delay constraint buffer constraint are satisfied In the following, we illustrate the use of dynamic service guarantees by a dynamic window-flow-control problem Example 67 (Dynamic Window Flow Control): Consider a network with the input the output Suppose that the network enforces a dynamic window flow control for the input with the dynamic window size We assume that For the dynamic window-flow-control system, the effective input to the network, denoted by, satisfies (41) Observe that, where is the function with for One may rewrite (41) as follows: Also, we assume that the network is a dynamic the effective input, ie, In conjunction with (42) (42) -server for (43) where we apply the distributive property the associativity of Since we assume that

11 CHANG et al: MIN, SYSTEM THEORY FOR CONSTRAINED TRAFFIC REGULATION AND DYNAMIC SERVICE GUARANTEES 815 We then have from Lemma 22(iii) that Thus, the dynamic window-flow-control system is a dynamic -server VII DYNAMIC SCED SCHEDULING ALGORITHM In this section, we define a scheduling algorithm, called the dynamic SCED algorithm, which we will show achieves the dynamic service guarantees in Section VI Consider a server with a time-varying capacity Let be the maximum number of packets that can be served at time, be the cumulative capacity in the interval A policy is called the EDF if the server schedules the packets according to their deadlines Note that the EDF policy is work conserving, ie, the server serves packets whenever there are packets at the server Now consider feeding streams of inputs to such a server Let be the cumulative number of packet arrivals of the th stream up to time Each packet is assigned a deadline We assume that the deadlines within the same stream are nondecreasing Also, let be the number of packets from the th stream that have deadlines not greater than As we assume the deadlines for each stream is nondecreasing, packet from stream is assigned the deadline from the following inverse mapping: (44) Theorem 71: Suppose that the server is operated under the EDF policy The necessary sufficient condition for every packet to be served not later than its deadline is ii) We prove the sufficient part by contradiction, as in [25] Suppose that the first packet that misses its deadline occurs at time Let be the last slot no later than such that the server serves less than packets Since the EDF policy is work conserving,, as there is at least one stream packet backlogged at time Moreover, there are exactly packets served in the interval Now let be the last slot in the interval, during which a packet with deadline greater than is served If all the packets served during the interval have deadlines less than or equal to, then define (in this case, there are no backlogged packets at the end of slot ) Thus, during the interval, exactly packets are served, each of these packets has a deadline that is less than or equal to Let be the set of streams that are not backlogged at the end of slot We claim that those packets served in can only come from the streams in Suppose that stream is not in Since there is a packet with deadline greater than that is served in slot, all the backlogged stream packets at the end of slot must have deadlines greater than This implies all the stream packets with deadlines not greater than have been served, as we assume the deadlines are nondecreasing within the same stream Thus, those packets served in can only come from the streams in, as those packets have deadlines less than or equal to Now suppose that stream is in As there are no backlogged stream packets at the end of slot, all the stream packets that arrive not later than have been served Thus, the number of stream packets that can be served in is bounded above by This, in turn, implies that the number of packets served in is bounded above by As there is a packet that misses its deadline at time, the bound is strict Thus, (45) for all for every that is a subset of Proof: i) We first prove the necessary part Let be the cumulative number of packet departures from all streams up to time Since the EDF policy is work conserving, we have from Example 34 that As we assume that every packet is served not later than its deadline for some that is a subset of with As is a subset of, we have a contradiction to (45) Lemma 72: Suppose we choose some, If for all, then all the packets are served not later than their deadlines Such a deadline assignment scheme is called the dynamic SCED algorithm in this paper Proof: It suffices to verify that the sufficient condition in Theorem 71 is satisfied Note that for every in (46) for all Let be a subset of Suppose that we now only have the streams in (the packets from other streams are discarded) Based on a stard sample path argument, it is clear that, under the EDF policy, every packet is still served not later than its deadline Following the same argument as in (46) yields (45)

12 816 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 10, NO 6, DECEMBER 2002 where we use in the last two inequalities The next lemma implies that deadlines in the dynamic SCED algorithm can be assigned in real time Specifically, if packet from stream arrives during slot, can be computed without knowledge of for Lemma 73: Suppose packet from stream arrives during slot Under the dynamic SCED algorithm, where Proof: Note that, under the dynamic SCED algorithm (47) (48) Since packet arrives at time,wehave for Thus, when, which implies that by definition of in (48) Therefore, by definition of,wehave By definition of, this implies To show the reverse inequality, note that by definition of, we have (49) Since for is nonnegative, inequality (49) implies that By definition of, this then implies that In Theorem 74, we state the admission criteria for the dynamic SCED algorithm for a server with a time-varying capacity Once the admission criteria are met, the dynamic SCED algorithm can then be used for providing dynamic service guarantees Theorem 74: A set of arrival streams, indexed, arrives to a server The arrival sequence of the th stream is denoted by, is known to conform to the dynamic upper envelope The server has a time-varying capacity to serve up to packets during slot Under the dynamic SCED algorithm, the server is a dynamic -server for for all if the following condition is satisfied for all : (50) Proof: As we assume that conforms to the dynamic upper envelope, we have from Lemma 32(ii) that Thus, where we apply the associativity of From Lemma 72, it then follows that all the packets are served before their deadlines Denote by the cumulative number of departures from stream by time Thus, the server is a dynamic -server for for all VIII CONCLUSIONS By extending the filtering theory under the ) algebra to the time-varying setting, we solved the problem of constrained traffic regulation For a constrained traffic-regulation problem with maximum tolerable delay maximum buffer size, we showed that the optimal regulator that generates the output traffic conforming to a dynamic envelope minimizes the number of discarded packets is a concatenation of the maximal dynamic -clipper with the maximal dynamic -regulator To provide dynamic service guarantees in a network, we developed the concept of the dynamic -server as a basic network element We showed that dynamic servers can be joined by concatenation, filter bank summation, feedback to form a composite dynamic server We also proposed the dynamic SCED scheduling algorithm to achieve dynamic service guarantees for a work-conserving link subject to multiple inputs One possible application of the time-varying filtering theory is dynamic admission control For a given connection, we may define a service curve to be guaranteed over the interval if a dynamic service curve is guaranteed, where if if if if if if For such a definition for dynamic service guarantees, an interesting problem is to find the relaxation time such that connection has virtually no impact on the admission criteria in Theorem 74 after Finally, we note that our approach is also applicable in the continuous-time setting, as shown in [21] [23] We also note that the bivariate function could be rom By specifying the probabilistic characteristics of the bivariate function,it is possible to provide probabilistic guarantees Previous results along this line could be found in [7] [13] REFERENCES [1] R Agrawal, R L Cruz, C Okino, R Rajan, Performance bounds for flow control protocols, IEEE Trans Networking, vol 7, pp , June 1999 [2] R Agrawal R Rajan, A general framework for analyzing schedulers regulators in integrated services network, in Proc 34th Annu Allerton Commun, Contr, Computing Conf, Oct 1996, pp [3], Open closed loop control in integrated services networks, in Proc 36th IEEE CDC, vol 2, Dec 1997, pp

13 CHANG et al: MIN, SYSTEM THEORY FOR CONSTRAINED TRAFFIC REGULATION AND DYNAMIC SERVICE GUARANTEES 817 [4] R Agrawal, F Baccelli, R Rajan, An algebra for queueing networks with time varying service its application to the analysis of integrated service networks, Elect Comput Eng Dept, Univ Wisconsin-Madison, Madison, WI, Tech Rep ECE-98-2 [Online] Available: May 1998 [5] F Baccelli, G Cohen, G Oslder, J Quadrat, Synchronization Linearity: An Algebra for Discrete Event Systems New York: Wiley, 1992 [6] C S Chang, Stability, queue length, delay of deterministic stochastic queueing networks, IEEE Trans Automat Contr, vol 39, pp , May 1994 [7], On the exponentiality of stochastic linear systems under the max-plus algebra, IEEE Trans Automat Contr, vol 41, pp , Aug 1996 [8], On deterministic traffic regulation service guarantees: A systematic approach by filtering, IEEE Trans Inform Theory, vol 44, pp , May 1998 [9], Matrix extensions of the filtering theory for deterministic traffic 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algorithm for VBR connections over a VBR trunk, in Proc ITC 15, Washington, DC, June 1997, pp [25] C L Liu J W Layl, Scheduling algorithms for multiprogramming in a hard-real-time environment, J Assoc Comput Mach, vol 20, pp 46 61, 1973 [26] F Lo Presti, Z-L Zhang, Z-L D, Z-L Towsley, J Kurose, Source time scale optimal buffer/bwidth trade-off for regulated traffic in an ATM node, in Proc IEEE Infocom 97, Kobe, Japan, pp [27] H Sariowan, R L Cruz, G C Polyzos, SCED: A generalized scheduling policy for guaranteeing quality-of-service, IEEE/ACM Trans Networking, vol 7, pp , Oct 1999 [28] A Skorokhod, Stochastic equations for diffusion processes in a bounded region, Theory Probab Appl, vol 6, pp , 1961 Cheng-Shang Chang (S 85 M 86 SM 93) received the BS degree from the National Taiwan University, Taipei, Taiwan, ROC, in 1983, the MS PhD degrees from Columbia University, New York, NY, in , respectively, all in electrical engineering From 1989 to 1993, he was a Research Staff Member with the IBM Thomas J Watson Research Center, Yorktown Heights, NY Since 1993, he has been with the Department of Electrical Engineering, National Tsing Hua University, Taiwan, ROC, where he is a Professor His current research interests are concerned with queueing theory, stochastic scheduling, performance evaluation of telecommunication networks parallel processing systems He authored Performance Guarantees in Communication Networks (New York: Springer, 2000) served as an Editor for Operations Research from 1992 to 1999 Dr Chang was the recipient of an IBM Outsting Innovation Award in 1992, an IBM Faculty Partnership Award in 2001, Outsting Research Awards from the National Science Council, Taiwan, R,OC, in , respectively Rene L Cruz (S 80 M 84 SM 90) received the BS PhD degrees from the University of Illinois at Urbana-Champaign, in , respectively, the SM degree from the Massachusetts Institute of Technology (MIT), Cambridge, in 1982, all in electrical engineering Since 1987, he has been with the University of California at San Diego, La Jolla, where he is currently a Professor of electrical computer engineering Dr Cruz was a program co-chair of the 2001 IEEE INFOCOM Conference a general co-chair of the Association for Computing Machinery (ACM) SIGCOMM Conference He was an associate editor for the IEEE TRANSACTIONS ON INFORMATION THEORY from 1994 to 1997 Jean-Yves Le Boudec graduated from the Ecole Normale Superieure de Saint-Cloud, Paris, France He received the Doctorate degree from the University of Rennes, Rennes, France, in 1984 In 1987, he joined Bell Northern Research, Ottawa, ON, Canada, where he was a Member of Scientific Staff with the Network Product Traffic Design Department In 1988, he joined the IBM Zurich Research Laboratory, Zurich, Switzerl, where he was Manager of the Customer Premises Network Department In 1994, he joined the professor at Ecole Polytechnic Fédérale de Lausanne (EPFL) Lausanne, Switzerl, as a Professor is currently a Full Professor His interests are in the architecture performance of communication systems Patrick Thiran (S 88 M 90) received the Electrical Engineering Degree from the Université Catholique de Louvain, Louvain-la-Neuve, Belgium, in 1989, the MS degree in electrical engineering from the University of California at Berkeley, in 1990, the PhD degree from the Ecole Polytechnic Fédérale de Lausanne (EPFL), Lausanne, Switzerl, in 1996 While with the EPFL, he was initially with the Circuits Systems Group, where he was involved with nonlinear circuits systems (including neural networks) until 1996 He then joined the Communication Networks Laboratory, EPFL, where he became a Professor (Professeur titulaire) in 1998 From 2000 to 2001, he was with Sprintlabs, Burlingame, CA He is currently an Assistant Professor with the School of Computer Communication Sciences, EPFL His research interests are communication networks, performance analysis, dynamical systems, stochastic models Dr Thiran is a Fellow of the Belgian American Educational Foundation He was an associate editor for the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS from 1997 to 1999 He was the recipient of the 1996 EPFL Doctoral Prize